Browsing by Author "Gergün, Seçil"

Now showing 1 - 3 of 3

Citation - Scopus: 1
Analysis on α-time scales and its applications to Cauchy-Euler equation
(Natural Sciences Publishing, 2024) Burcu Silindir; Seçil Gergün; Ahmet Yantir; Gergün, Seçil; Silindir, Burcu; Yantir, Ahmet
This article is devoted to present the α-power function calculus on α-time scale the α-logarithm and their applications on α-difference equations. We introduce the α-power function as an absolutely convergent infinite product. We state that the α-power function verifies the fundamentals of α-time scale and adheres to both the additivity and the power rule for α-derivative. Next we propose an α-analogue of Cauchy-Euler equation whose coefficient functions are α-polynomials and then construct its solution in terms of α-power function. As illustration we present examples of the second order α-Cauchy-Euler equation. Consequently we construct α-analogue of logarithm function which is determined in terms of α-integral. Finally we propose a second order BVP for α-Cauchy-Euler equation with two point unmixed boundary conditions and compute its solution by the use of Green’s function. © 2024 Elsevier B.V. All rights reserved.
Citation - WoS: 3
Citation - Scopus: 4
Generalized Polynomials and Their Unification and Extension to Discrete Calculus
(MDPI, 2023) Mieczyslaw Cichon; Burcu Silindir; Ahmet Yantir; Secil Gergun; Silindir, Burcu; Yantir, Ahmet; Cichoń, Mieczysław; Gergün, Seçil
In this paper we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q,h)-time scales. Our proposed approach encompasses both difference and quantum problems making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator alpha that combines both jump operators in a convex manner. This allows us to generate a time scale alpha which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale alpha which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover we create some polynomials on alpha-time scales using the alpha-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally we establish the alpha-version of the Taylor formula. Finally we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus.
Citation - WoS: 3
Citation - Scopus: 3
Power function and binomial series on T(q-h)
(TAYLOR & FRANCIS LTD, 2023) Secil Gergun; Burcu Silindir; Ahmet Yantir; Gergün, Seçil; Silindir, Burcu; Yantir, Ahmet
This article is devoted to present (q h) -analogue of power function which satisfies additivity and derivative properties similar to the ordinary power function. In the light of nabla (q h) -power function we present (q h)-analogue of binomial series and conclude that such power function is (q h)-analytic. We prove the analyticity by showing that both the power function and its absolutely convergent Taylor series solve the same IVP. Finally we present the reductions of (q h)-binomial series to classical binomial series Gauss' binomial and Newton's binomial formulas.